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It depends on the density of the oil in relation to the density of the object. Any object will sink in any fluid until the density of the object and the fluid reach equilibrium.

The density of a fluid increases as the depth of the fluid increases. Water at the surface has a density of 1 gram per cubic centimeter. At 100 meters depth the pressure acting on the water with no change in temperature equals 10kg/square cm.
This compression on the water increases its density per cubic centimeter to about 1.000464 grams per cubic centimeter. This isn't the same with all liquids, some are more or less compressible than water. An incompressible object with a density of 1.000464 grams per cubic centimeter, however, would sink in the water until it reaches somewhere about 100 meters depth. Then it would float there. [The density changes with temperature as well, so that a slight change in temperature would make the object more or less buoyant at that depth.] This tendency to find equilibrium is the principle upon which submarines are able to operate, changing depth by increasing or decreasing their relative density.

An object with 8 grams per liter density will float in most liquids. An object with a density of 8 grams per cc will sink in most liquids.

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